### Abstract

Stochastic Ising and voter models on ℤ^{d} are natural examples of Markov processes with compact state spaces. When the initial state is chosen uniformly at random, can it happen that the distribution at time t has multiple (subsequence) limits as t → ∞? Yes for the d = 1 Voter Model with Random Rates (VMRR) - which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature - if the disorder distribution is heavy-tailed. No (at least in a weak sense) for the VMRR when the tail is light or d ≥ 2. These results are based on an analysis of the "localization" properties of Random Walks with Random Rates.

Original language | English (US) |
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Pages (from-to) | 417-443 |

Number of pages | 27 |

Journal | Probability Theory and Related Fields |

Volume | 115 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1999 |

### Keywords

- Chaotic time dependence
- Disordered spin system
- Random environment
- Random walk
- Stochastic Ising model
- Voter model

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Fontes, L. R. G., Isopi, M., & Newman, C. M. (1999). Chaotic time dependence in a disordered spin system.

*Probability Theory and Related Fields*,*115*(3), 417-443. https://doi.org/10.1007/s004400050244